Matched Metrics to the Binary Asymmetric Channels

Abstract

In this paper we establish some criteria to decide when a discrete memoryless channel admits a metric in such a way that the maximum likelihood decoding coincides with the nearest neighbor decoding. In particular we prove a conjecture presented by M. Firer and J. L. Walker establishing that every binary asymmetric channel admits a matched metric. Index TermsBinary asymmetric channel, channel model, maximum likelihood decoding I. INTRODUCTIONAs it is well known, maximum likelihood decoding (MLD) over a symmetric channel coincides with nearest neighbor decoding (NND) with respect to the Hamming metric. In this paper we deal with the problem of matching a metric to a given channel regarding the decoding criteria mentioned above. This problem was considered in 1967 by J. L. Massey [2] where a metric matched to a discrete memoryless channel is defined as a metric for which the NND is a MLD.Since then, this type of matching have been studied in some special cases. For instance, certain channels matching to the Lee metric were obtained in [1]. The problem of matching a metric to a channel was taken up by G. Séguin [3], where the main focus was on sequences of additive metrics. In the referred paper it is used a stronger condition also assumed here: a metric matched to a channel is one for which not only the NND is a MLD but also the MLD is a NND. The

Parameters of Codes for the Binary Asymmetric Channel

“…Shortly after that, Qureshi showed that the binary asymmetric channel admits a matched metric [10], establishing a conjecture of Firer and Walker [6]. The approach of [10] develops criteria (and algorithms) to understand when a discrete memoryless channel admits a metric for which the maximum likelihood decoder coincides with the nearest neighbour decoder. A current open problem is to explicitly describe the metrics matching the binary asymmetric channel.…”

“…In 2016 it was proved in [9] that the binary asymmetric channel is metrizable in the weak sense of Massey [8]. Shortly after that, Qureshi showed that the binary asymmetric channel admits a matched metric [10], establishing a conjecture of Firer and Walker [6]. The approach of [10] develops criteria (and algorithms) to understand when a discrete memoryless channel admits a metric for which the maximum likelihood decoder coincides with the nearest neighbour decoder.…”

Optimal Combinatorial Neural Codes with Matched Metric $δ_{r}$: Characterization and Constructions

“…Further researches on "Matched " metric Figure 2. Binary asymmetric memoryless channel ( [13], [19], [20]) to focus the ability on the usual decoding algorithms (the maximum likehood decoding and nearest neighbour decoding ) for CN codes. In [6], the authors suggested and discussed two kinds of metrics for CN codes.…”